Questions tagged [topology]
In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.
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Conditions for Bogoliubov-de Gennes Hamiltonian representation
The $H_{BdG}$ hamiltonian is described in topocondmat.org as follows:
here we can see that the submatrices along the diagonal are related as negative of complex conjugate of each other, I feel that ...
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Does physicists have a pre-conceived notion of continuity?
In many physics lecture on GR/ mathematical physics, one of the first things discussed is topology. I have seen many times that the reason for topology being discussed is that it's the weakest ...
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Will a light come back within finite years?
In this answer Javier said
Imagine the universe was the inside of a ball. We're 3D now, so no one is hiding any dimensions. This ball has a border, except it's not really a border. You should think ...
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Could the universe be a 4-ball?
I recently thought of the idea that the universe could be an infinite 4-ball. The Big Bang would be its centre, and time would be outward from its centre (one layer would be one point in time). I ...
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Can space only be infinite? [closed]
I have read before that if you could just go fast enough, as a thought experiment, and you move in a straight line, in any direction, that you eventually might reach the spot from which you started. I ...
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Reduction of the gravity gauge from various groups
As a gauge theory, the classic reduction for gravity is from the frame bundle to the Lorentz group,
\begin{equation}
GL(4) \to O(3,1)
\end{equation}
The associated configuration space of that ...
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What are the surfaces that contain an interior volume (space separating) called? Are they related to orientability?
I know that a "closed" surface is defined as a compact surface with no boundary. I don't have it clear if they have something to do with having an interior volume (completely enclosed volume)...
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Mathematics of the star-mesh transformation
I'm trying to understand the star-mesh transform from a mathematical perspective. This transformation removes a node and by definition, the topology of the network changes, but the resulting network ...
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What is a causal structure?
The notion of "causal structure" brings up many different notions in general relativity.
It is associated with the fiber bundle $\pi : \mathcal{C} \to M$ set of every light cone at every ...
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What is the correct domain of integration for the index of instantons? - $\mathbb{R}^4$ or $S^4$?
I posted the original question on Math SE but it seems like a more appropriate question for Physics SE:
https://math.stackexchange.com/q/4417225/
In calculating the instanton solutions for $SU(2)$ ...
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Photons as Observers and the Extended Real Number Line Topology
I am not a physicist. This is the first question I write in such a forum so if there are remarks on how I wrote it, I'll be happy to edit.
I am originally a mathematician with some interest in physics....
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AdS$_4$ and $\mathbb{H}^4$: What is the difference between them?
This figure (source) shows the embedding of 4D hyperbolic space $\mathbb{H}^4$ and 4D de Sitter space dS$_4$ in 5D Minkowski space $\mathbb{M}^5$. $\mathbb{H}^4$ is a hyperboloid of two sheets and dS$...
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Emergent spacetime from a web of string world sheets?
Like many, I have deep conceptual difficulty understanding how an enormous amount of closed strings can become an effective classical spacetime that satisfies Einstein's equations. I appreciate that ...
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Quadruple index and the existence of corner state
I have been following this paper, which seems to discuss the connection between corner state and a quadruple index calculated $q(k_z)$ by equation (3) of it. At two special point $k_z=0 /\pi$, the ...
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Penrose diagrams for non-spherically symmetric spacetimes
As far as I have seen, Penrose diagrams are composed for spacetimes where there is spherical symmetry. The angular degrees of freedom are suppressed so as to understand the causal properties of ...
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How to calculate the variation of the metric on a compact manifold?
For example, given a torus with a modular parameter $\tau$ and an action
\begin{equation}
I=\frac{g}{2}\int_\mathcal{M} d^2 z \sqrt{-g}\ g_{ij}(z) \partial^i\phi \partial^j\phi
\end{equation}
...
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Does gap closing and reopening guarantee a non-trivial topological phase?
I know that a Dirac point carries a Chern number of $\pm\frac{1}{2}$ and when we have a gap closure at any point in our band structure we can transfer Chern numbers depending on at how many points we ...
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Topological classification of the Kane-Mele model?
Where should the Kane-Mele Model fall in the 10-fold way topological classification? I see that it is on a honeycomb lattice which is bipartite and thus has particle-hole symmetry. Going by that ...
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Quantum Field Theory on non-globally hyperbolic spacetimes?
In all the references I have found on QFT in curved spacetime, they treat only globally hyperbolic Lorentzian spacetimes, not Lorentzian spacetimes in general. Are there any references which discuss ...
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Obtaining Dirac spectrum on unorientable manifold ($RP^n$) from orientable manifold
The Dirac spectrum for $S^n$ is well known along with its multiplicities. In Appendix D of https://arxiv.org/abs/1510.05663 author computes dirac spectrum of $RP^4$ from that of $S^4$. The argument ...
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Any problem creating a global time function given known spacetime topology?
Gödel says the absence of a global time function
seems to imply an absurdity. For it enables one e.g., to travel into the near past of those places where he has himself
lived.
But is it not true ...
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Helicity for massless particles
The little group for massless particles is $ISO(2)$, with the following Lie algebra: $$[A,B]=0, \; [J^3,A]=iB, \; [J^3,B]=-iA,$$ where $A,B$ generate translations and $J^3$ generates rotations. To ...
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Stability of deformed planets
Motivated by this question and first line of this answer, I want to ask if we were describe the evolution of an orientable continuous simply connected physical body having mass distribution based on ...
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Misconception about closed string worldsheet definition
I'm a little confused about the precise way to define the worldsheet $\Sigma$ of a closed string. Its parametrization must be of the form $X: \Sigma \longrightarrow \mathbb{R}^{1,D-1}$ and one of its ...
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How to find the deficit angle of the BTZ black hole?
I've read the the answer to this question where it is claimed that the BTZ black hole has a deficit angle. How can we see that in the BTZ case and in general?
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Laughlin gauge argument, integer quantum hall and periodic boundary conditions
In every treatment I have seen of the Laughlin gauge argument, it is suggested that as a flux quantum, $\Phi_0 = h/e$, threads through the cylinder or ribbon, that one unit of charge is pumped from ...
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(Equivalent theory of) general relativity without general relativity language
Is there a formulation or theory of classical relativistic gravity yielding the same predictions as the standard General Relativity (when the predictions are expressed in GR-free language which ...
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Interactions in String Theory
I study in Szabo's book and in the "String Perturbation Theory" section he says the following:
in string theory the structure of interactions is completely determined by the free worldsheet ...
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More rigorous explanation as to why $G/H = G'/H'$ for vacuum manifolds?
When studying topological defects, the parameter space (or vacuum manifold in QFT) is denoted as the coset space $G/H$, where $G$ corresponds to the symmetry group of the Lagrangian and $H$ the ...
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Quantization of Helicity for massless particles [duplicate]
My understanding is that the quantization of Helicity for little group of massive particles comes from the fact that rotation in space leaves the 4-momentum $P^\mu=(m,0,0,0)$ invariant; we know that $...
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Homotopy group of $[SU(2) \times U(1)] / U(1)$ dependent on embedding of $U(1)$?
I have read, that the topology and thus the homotopy groups of $[SU(2) \times U(1)] / U(1)$ depend upon the embedding of $U(1)$ into $SU(2) \times U(1)$.
For the Electroweak theory inside the standard ...
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Energy density of a cosmic string in the Abelian Higgs Model (Nielsen-Olesen Vortex)
my question is what the energy density of a cosmic string in the abelian higgs model is? I have different sources (Scholarpedia Cosmic Strings, the book "Cosmic Strings and Topological Defects&...
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Does physical prediction depend on the topologies that we put in the space of smooth section of a vector bundle?
In the artice Properties of field functionals and characterization of local functionals, we have
Let $M$ be a manifold and $B \rightarrow M$ be a smooth vector bundle of $\operatorname{rank} r$ over $...
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What is a Taub–NUT space?
I trying to read and understand the Taub–NUT space. Wikipedia introduced that it is associated with the Taub–NUT metric, an exact solution to Einstein's equations. However, I didn't find a direct ...
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Can every canonical transformation be broken down into a large number of infinitesimal canonical transformations?
Consider a canonical transformation from $(q,p)$ to $(Q,P)$ depending upon a continuous parameter $\alpha$ such that:
$$Q_i=Q_i(q,p,t,\alpha), \space P_i=P_i(q,p,t,\alpha)$$
where $q$ and $p$ ...
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What do the '4' and 'b' signify in a layer of a crystal called a '4Hb' crystal or material?
From Phys.org:
Study gathers evidence of topological superconductivity in the transition metal 4Hb-TaS2
Which, in turn, references:
Abhay Kumar Nayak et al, Evidence of topological boundary modes with ...
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How to determine the distance between any two sites of a finite lattice subjected to periodic boundary conditions?
I want to study the Ising model on a finite kagome lattice assuming periodic boundary conditions (PBC) and long range interactions. More specifically, all spin pairs contribute to the total energy, so ...
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Literature request: Non-compact supported gauge transformations
I have recently heard that non-compact supported gauge transformations of non-abelian gauge theories have a non-trivial effect on the Hilbert space of states.
This is a topic I can not find anywhere ...
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Why is the QCD $\theta$-term aware of the topology of the space?
The QCD Lagrangian without the $\theta$-term $$\mathcal{L}_{QCD}=-\frac{1}{4}G_{\mu\nu}^aG^{\mu\nu a}\tag{1}$$ is not topological. However, the $\theta$-term $$\mathcal{L}_\theta=\frac{\theta}{32\pi^...
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Moduli space for Riemann surfaces with boundaries and open string loop diagrams
I'm searching for information on the moduli space for Riemann surfaces with boundaries, like the ones used to compute open string loop diagrams. I found a huge lot of info for the case without ...
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Discrete conformal transformation of $\mathbb{R}^{p,q}$
We know the flat space $\mathbb{R}^{p,q}$ with the metric $ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu$ can be conformally compactified to the quadric $-{\xi^0}^2 + \eta_{\mu\nu} \xi^\mu \xi^\nu + {\xi^d}^2 = ...
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Do physical systems have intrinsic degrees of freedom that are independent of its representation?
Considering just the Newtonian case, suppose we have a system described by $n$ canonical position-momentum pairs, $(p_1,q_1),\dots,(p_n,q_n)$, and a Hamiltonian $H$. If we "scrubbed" all the ...
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Show that this expanding cylinder will unravel this loop
Below is a really hard physics(?) problem which I honestly don't even know how to begin. I thought of it when reading a knot theory book.
Below in the picture, you see the projection of an unknot (a ...
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Is the phenomenon of geometrical frustration in condensed matter physics related to some kind of topological invariant?
Edit (attempt to clarify my question a little bit):
I’m not thinking geometrical frustration should be necessarily associated to a topological invariant in a direct way, but maybe local geometrical ...
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Carroll's Spacetime and Geometry - Notion of open subset of a manifold
In Sean Carroll's Spacetime and Geometry, an introductory section on manifolds contains the following:
A chart or coordinate system consists of a subset $U$ of a set $M$ along with
a one-to-one map $\...
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Cosmic strings, domain walls, and magnetic monopoles are topological defects, but what's defecting?
Wikipedia gives an explanation of cosmic strings that I'm sure would be very helpful if I had a major in topology, but alas I do not. I know that a topological defect is any sort of discontinuity in a ...
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Kitaev chain and fermionic parity
I am learning the free open course on topological insulator on edX.
And I am reading the section: Bulk-edge correspondence in Kitaev Chain. I am stuck at the subsection: Connecting the bulk invariant ...
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Geodesically complete spacetime
By definition, a spacetime is geodesically complete if all inextendible curve are complete but is this equivalent to «if all geodesic of finite length has endpoints»?
My situation is:
I have a set of ...
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Running electrolysis on inside surface of a tube
I use electrolysis for rust removal and electroplating. I notice that when one of the electrodes is a closed tube or has other topological holes with a sufficiently large height:diameter ratio, the ...
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Euclidean space to Minkowski spacetime
Can you continuously deform (i.e., shrink, twist, stretch, etc. in any way without tearing) four-dimensional Euclidean space to make it four-dimensional Minkowski spacetime?
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